# Multinomial Goodness-of-Fit Based on U-Statistics: High-Dimensional Asymptotic and Minimax Optimality

## TL;DR

This paper develops a new family of U-statistics for multinomial goodness-of-fit tests in high-dimensional settings, addressing limitations of traditional tests like Pearson's chi-squared, and establishes their asymptotic properties and optimality.

## Contribution

It introduces a novel class of U-statistics for high-dimensional multinomial goodness-of-fit testing with proven asymptotic distributions and minimax optimality.

## Key findings

- U-statistics are asymptotically Poisson or Gaussian.
- The proposed tests achieve minimax rate optimality.
- Traditional chi-squared tests have reduced power in high dimensions.

## Abstract

We consider multinomial goodness-of-fit tests in the high-dimensional regime where the number of bins increases with the sample size. In this regime, Pearson's chi-squared test can suffer from low power due to the substantial bias as well as high variance of its statistic. To resolve these issues, we introduce a family of U-statistic for multinomial goodness-of-fit and study their asymptotic behaviors in high-dimensions. Specifically, we establish conditions under which the considered U-statistic is asymptotically Poisson or Gaussian, and investigate its power function under each asymptotic regime. Furthermore, we introduce a class of weights for the U-statistic that results in minimax rate optimal tests.